Brendan Rodgers' side swept aside their SSE Airtricity League opponents with a confident display from their second team
Iago Aspas opened the scoring early on, with Fabio Borini, Jack Dunn and Martin Kelly all adding to the tally in the second period as the Reds ran out comfortable winners.
Liverpool controlled the tempo of the game and had the majority of the possession as expected, but Shamrock Rovers matched them in the first half and were unlucky not to have a goal to show for their efforts.
It was Aspas opened the scoring for the visitors when Jordon Ibe cut a fine ball back to him, allowing the midfielder to finish with ease, sweeping the ball past keeper Barry Murphy and into the net.
The Hoops defended well to withstand the Liverpool attack, looking to hit their Premier League opponents on the break, with Ciaran Kilduff and Ryan Brennan going close with two fine headers.
But Borini scored on his Liverpool return when he fired a low driven shot beyond Barry Murphy at the start of the second half to double his side's lead and Dunn and Kelly sealed the victory for the Reds.